Complex Numbers - Absolute Values
The absolute value of a number is often viewed as the "distance" a number is away from 0, the origin.
Contents
General Concepts
For real numbers, the absolute value is just the magnitude of the number without considering its sign. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.
For a complex number \(z = a + bi\) represented on the complex plane by the pair \((a, b)\), the "distance" from the origin is found using the Pythagorean theorem. The absolute value of \(z\) is defined as
\[|a+bi| = \sqrt{a^2 +b^2}.\]
For example, the absolute value of the complex number \(3+4i\) is equal to
\[|3+4i| = \sqrt{3^2 +4^2} = 5.\]
The absolute value can also be written as
\[|z| = \sqrt{z \bar{z}},\]
where\(\bar{z}\) is the complex conjugate of \(z.\)
Examples/Problems
What is the absolute value of the complex number \(-5+12i?\)
We have \(\lvert -5+12i \rvert=\sqrt{(-5)^2+12^2}=\sqrt{25+144}=\sqrt{169}=\sqrt{13^2}=13.\) \( _\square\)
The absolute value of the complex number \(7+bi\) is \(\sqrt{170}.\) What is the negative number \(b?\)
The absolute value of \(7+bi\) is \(\lvert 7+bi \rvert =\sqrt{7^2+b^2}=\sqrt{170},\) implying \(7^2+b^2=170,\) or \(b^2=121.\) This implies \(b=\pm 11.\) Since \(b\) is negative, \(b=-11.\) \( _\square\)
The absolute value of the complex number \(-7-8i\) can be expressed as \[\lvert -7-8i \rvert=\sqrt{(-7-8i)(a+bi)},\] where \(a\) and \(b\) are real numbers and \(i\) is the imaginary number. What are \(a\) and \(b?\)
Observe that the absolute value of a complex number \(z\) can be written as \(\lvert z \rvert = \sqrt{z \bar{z}},\) where\(\bar{z}\) is the complex conjugate of \(z.\) Then since \(z=-7-8i,\) thus \(a+bi=-7+8i,\) which implies that \(a=-7\) and \(b=8.\) \( _\square\)
What is the absolute value of the following sum of complex numbers: \[(12+4i)-(9-13i)-3i?\]
We have \(\lvert (12+4i)-(9-13i)-3i \rvert = \lvert 3+14i \rvert =\sqrt{3^2+14^2}=\sqrt{205}.\) \( _\square\)
Let \((6, 8)\) be the coordinates of the complex number \(z\) on the complex plane. Then what is the absolute value of its complex conjugate \(\bar{z}?\)
Since \(z=6+8i,\) it follows that \(\bar{z}=6-8i.\) Then \[\lvert \bar{z} \rvert=\lvert6-8i \rvert=\sqrt{(6)^2+(-8)^2}=\sqrt{36+64}=\sqrt{100}=\sqrt{10^2}=10. \ _\square\]
The complex number \(a+bi\) has the same absolute value as a different complex number \(3+4i.\) If \(a\) and \(b\) are both positive integers, what are \(a\) and \(b?\)
The absolute value of \(3+4i\) is \(\lvert 3+4i \rvert =\sqrt{3^2+4^2}=\sqrt{25}=5.\) Since \(a+bi\) also has an absolute value of \(5,\) it follows that \(\lvert a+bi \rvert =\sqrt{a^2+b^2}=5,\) implying \(a^2+b^2=25.\) Since \(a\) and \(b\) are both positive integers and \(a+bi\ne 3+4i\) by assumption, the only way that \(a^2+b^2=25\) holds is that \(a=4\) and \(b=3.\) \( _\square\)
Let \(x_1,x_2\) be the roots of the quadratic equation \(x^2+ax+b=0,\) where \(a\) and \(b\) are complex numbers, and \(y_1,y_2\) are the roots of the equation \(y^2+| a |y+| b|=0 \).
If \(| x_1 |=| x_2 |=1,\) what is \( | y_1|+| y_2 |= ? \)
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Note: \(| \cdot |\) denotes the absolute value function.